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1
Content available The discrete, fractional order model of a two-dimensional temperature field using Grünwald-Letnikov definition
Oprzędkiewicz Krzysztof
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2024
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Vol. 72, nr 4
art. no. e150332
EN
In the paper a new, fractional order, discrete model of a two-dimensional temperature field is addressed. The proposed model uses Grünwald-Letnikov definition of the fractional operator. Such a model has not been proposed yet. Elementary properties of the model: practical stability, accuracy and convergence are analysed. Analytical conditions of stability and convergence are proposed and they allow to estimate the orders of the model. Theoretical considerations are validated using exprimental data obtained with the use of a thermal imaging camera. Results of analysis supported by experiments point that the proposed model assures good accuracy and convergence for low order and relatively short memory length.
2
Content available remote "Fractionalization" A new approach for comparing different approximation methods of fractional order systems and disturbances rejection in PID control
Dib Riad, Bensafia Yassine, Khettab Khatir
Przegląd Elektrotechniczny
|
2023
|
R. 99, nr 7
227-231
EN
Recently, many research works have focused on fractional order systems and their approximation methods. It has been shown to be a useful tool for enhancing plant dynamics in terms of time and frequency performance. In this paper we propose a new approach for comparing between the different approximations methods of fractional order systems and disturbance rejection in PID control of DC motor by fractionalizing an integer order derivative operator in the original integer system. The implementation of the fractionalized terms is realized by mean of the well established approximation methods and in order to determine the best method, the responses of original integer system are compared to those of fractionalized systems. Illustrative simulations examples show that the fractionalization approach give the best decision (selected method) ,a good tool for comparison between different approximation methods and it give the good rejection of disturbances in PID control of DC motor . This approach can also be generalized to others numerical approximation methods and it can also be used in the area of systems control.
PL
Ostatnio wiele prac badawczych koncentrowało się na systemach rzędu ułamkowego i metodach ich aproksymacji. Wykazano, że jest to przydatne narzędzie do zwiększania dynamiki instalacji pod względem wydajności czasowej i częstotliwościowej. W tym artykule proponujemy nowe podejście do porównywania różnych metod aproksymacji systemów ułamkowego rzędu i odrzucania zakłóceń w sterowaniu PID silnika prądu stałego poprzez frakcjonowanie operatora pochodnej rzędu całkowitego w oryginalnym układzie całkowitym. Implementacja wyrazów ułamkowych jest realizowana za pomocą dobrze znanych metod aproksymacyjnych iw celu wyznaczenia najlepszej metody porównuje się odpowiedzi oryginalnego układu całkowitoliczbowego z odpowiedziami układów ułamkowych. Ilustracyjne przykłady symulacyjne pokazują, że podejście frakcyjne daje najlepszą decyzję (wybrana metoda), jest dobrym narzędziem do porównywania różnych metod aproksymacyjnych i zapewnia dobre odrzucanie zakłóceń w regulacji PID silnika prądu stałego. Podejście to można również uogólnić na inne metody aproksymacji numerycznej, a także można je stosować w obszarze sterowania systemami.
3
Content available Fractional order, state space model of the temperature field in the pcb plate
Oprzędkiewicz Krzysztof, Mitkowski Wojciech, Rosół Maciej
Acta Mechanica et Automatica
|
2023
|
Vol. 17, no. 2
180--187
EN
In the paper the fractional order, state space model of a temperature field in a two-dimensional metallic surface is addressed. The proposed model is the two dimensional generalization of the one dimensional, fractional order, state space of model of the heat transfer process. It uses fractional derivatives along time and length. The proposed model assures better accuracy with lower order than models using integer order derivatives. Elementary properties of the proposed model are analysed. Theoretical results are experimentally verifed using data from industrial thermal camera.
4
Content available Controllability of linear convex combination of linear discrete-time fractional systems
Kaczorek Tadeusz, Klamka Jerzy, Dzieliński Andrzej
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2022
|
Vol. 70, nr 5
art. no. e143102
EN
In this paper the controllability properties of the convex linear combination of fractional, linear, discrete-time systems are characterized and investigated. The notions of linear convex combination and controllability in the context of fractional-order systems are recalled. Then, the controllability property of such a linear combination of discrete-time, linear fractional systems is proven. Further, the reduction of an infinite problem of transition matrix derivation is reduced to a finite one, which greatly simplifies the numerical burden of the controllability issue. Examples of controllable and uncontrollable, single-input, linear systems are presented. The possibility of extension of the considerations to multi-input systems is shown.
5
Content available Fractional order, discrete model of heat transfer process using time and spatial Grünwald-Letnikov operator
Oprzędkiewicz Krzysztof
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2021
|
Vol. 69, nr 1
art. no. e135843
EN
In the paper a new, state space, fully discrete, fractional model of a heat transfer process in one dimensional body is addressed. The proposed model derives directly from fractional heat transfer equation. It employes the discrete Grünwald-Letnikov operator to express the fractional order differences along both coordinates: time and space. The practical stability and numerical complexity of the model are analysed. Theoretical results are verified using experimental data.
6
Content available An output sensitivity problem for a class of fractional order discrete-time linear systems
Benfatah Youssef, El-Bhih Amine, Rachik Mostafa, Lafif Marouane
Acta Mechanica et Automatica
|
2021
|
Vol. 15, no. 4
227--235
EN
Consider the linear discrete-time fractional order systems with uncertainty on the initial state {Δαxi+1=Axi+Bui, i≥0x0=τ0+τ̂0∈Rn, τ̂0∈Ωyi=Cxi, i≥0}, where A,B and C are appropriate matrices, x0 is the initial state, yi is the signal output, α the order of the derivative, τ0 and τ̂0 are the known and unknown part of x0, respectively, ui=Kxi is feedback control and Ω⊂Rn is a polytope convex of vertices w1,w2,...,wp. According to the Krein–Milman theorem, we suppose that τ̂0=Σ pj=1αjwj for some unknown coefficients α1≥0,...,αp≥0 such that Σ pj=1αj=1. In this paper, the fractional derivative is defined in the Grünwald–Letnikov sense. We investigate the charac-terisation of the set χ(τ̂0,ϵ) of all possible gain matrix K that makes the system insensitive to the unknown part τ̂0, which means χ(τ̂0,ϵ)={K∈Rm×n / ∥∂yi∂αj∥≤ϵ, ∀j=1,...,p,∀i≥0}, where the inequality ∥∂yi∂αj∥≤ϵ showing the sensitivity of yi relative-ly to uncertainties {αj}j=1p will not achieve the specified threshold ϵ>0. We establish, under certain hypothesis, the finite determination of χ(τ̂0,ϵ) and we propose an algorithmic approach to made explicit characterisation of such set.
7
Content available remote Integer order vs fractional order temperature models in the forced air heating system
Oprzędkiewicz Krzysztof, Podsiadło Maciej, Dziedzic Klaudia
Przegląd Elektrotechniczny
|
2019
|
R. 95, nr 11
35--40
EN
The paper presents a comparing integer order (IO) vs fractional order (FO) transfer function models of the forced air heating system applied in housing heating. The proposed FO models are simpler to identify than analogical IO models and their accuracy in the sense of fitting to experimental results is practically the same. Results are verified using experimental data from the real isothermal room, used in a pharmaceutical factory.
PL
W pracy zaprezentowano porównanie modeli transmitancyjnych całkowitego i niecałkowitego rz˛edu dla systemu nagrzewu powietrza stosowanego w ogrzewaniu budynków. Proponowane modele sa˛ tak samo dokładne jak modele całkowitego rze˛du, a ich identyfikacja jest prostsza ze wzgle˛du na mniejsza˛ liczbe˛ parametrów. Prezentowane wyniki zostały zweryfikowane z wykorzystaniem danych eksperymentalnych z pomieszczenia izotermicznego stosowanego w fabryce farmaceutycznej.
8
Content available The Caputo vs. Caputo-Fabrizio operators in modeling of heat transfer process
Oprzędkiewicz K., Mitkowski W., Gawin E., Dziedzic K.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 4
501--507
EN
In the paper two non-integer order, state space models of heat transfer process are compared. The first uses a known Caputo operator and the second – a new operator proposed by Caputo and Fabrizio in 2015. Both discussed models are modifications of a known, integer order, state space, semigroup model of heat transfer process. Parameters of both models were identified by means of optimization of MSE cost function with the use of simplex method, available in MATLAB. Both proposed models have been compared in the aspect of accuracy and convergence. Analytical and numerical results show that the Caputo-Fabrizio model is faster convergent and easier to implement than the Caputo model. However, its accuracy in the sense of MSE cost function is worse.
9
Układy niecałkowitego rzędu i ich wybrane zastosowania
Mitkowski W.
Hutnik, Wiadomości Hutnicze
|
2014
|
Vol. 81, nr 1
39--43
PL
W pracy omówiono podstawowe własności układów niecałkowitego rzędu. Podano wybrane przykłady zastosowań z następujących obszarów: modelowanie, identyfikacja, regulacja automatyczna. Przedstawiono również przykłady symulacji komputerowych.
EN
The paper discusses the basic properties of fractional systems. Given some examples of applications of the following areas: modeling, identification, automatic control. It also presents examples of computer simulations.
10
Content available Approximation method for a fractional order transfer function with zero and pole
Oprzędkiewicz K.
Archives of Control Sciences
|
2014
|
Vol. 24, no. 4
447--463
EN
The paper presents an approximation method for elementary fractional order transfer function containing both pole and zero. This class of transfer functions can be applied for example to build model - based special control algorithms. The proposed method bases on Charef approximation. The problem of cancelation pole by zero with useful conditions was considered, the accuracy discussion with the use of interval approach was done also. Results were depicted by examples.
11
Content available Design of unknown input fractional-order observers for fractional-order systems
N’Doye I., Darouach M., Voos H., Zasadzinski M.
International Journal of Applied Mathematics and Computer Science
|
2013
|
Vol. 23, no. 3
491--500
EN
This paper considers a method of designing fractional-order observers for continuous-time linear fractional-order systems with unknown inputs. Conditions for the existence of these observers are given. Sufficient conditions for the asymptotical stability of fractional-order observer errors with the fractional order α satisfying 0 < α < 2 are derived in terms of linear matrix inequalities. Two numerical examples are given to demonstrate the applicability of the proposed approach, where the fractional order α belongs to 1 ≤ α < 2 and 0 < α ≤1, respectively. A stability analysis of the fractional-order error system is made and it is shown that the fractional-order observers are as stable as their integer order counterpart and guarantee better convergence of the estimation error.
12
Content available Fixed final time and free final state optimal control problem for fractional dynamic systems – linear quadratic discrete-time case
Dzieliński A., Czyronis P. M.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2013
|
Vol. 61, nr 3
681--690
EN
The optimization problem for fractional discrete-time systems with a quadratic performance index has been formulated and solved. The case of fixed final time and a free final state has been considered. A method for numerical computation of optimization problems has been presented. The presented method is a generalization of the well-known method for discrete-time systems of integer order. The efficiency of the method has been demonstrated on numerical examples and illustrated by graphs. Graphs also show the differences between the fractional and classical (standard) systems theory. Results for other cases of the fractional system order (coefficient ) and not illustrated with numerical examples have been obtained through a computer algorithm written for this purpose.
13
Content available Obserwatory dodatnich liniowych układów dyskretnych ułamkowego rzędu
Kociszewski R.
Pomiary Automatyka Robotyka
|
2010
|
R. 14, nr 2
492-500
PL
W pracy rozpatrzono problem syntezy obserwatorów dodatnich liniowych układów dyskretnych ułamkowego rzędu. Wykorzystując podejście oparte na liniowych nierównościach macierzowych (LMI) sformułowano warunki istnienia dodatniego asymptotycznie stabilnego obserwatora. Podano warunki konieczne i wystarczające obserwowalności dodatnich dyskretnych układów ułamkowego rzędu. Rozważania zilustrowano przykładem symulacyjnym.
EN
The paper is devoted to observer synthesis for linear positive fractional discrete-time systems. The problem of finding a nonnegative gain matrix of the observer such that the observer is positive and asymptotically stable is formulated and solved by the use of LMI method. Necessary and sufficient conditions for the observability for linear positive fractional discrete-time systems are formulated and proved. The proposed approach to computation the nonnegative gain matrix is illustrated by numerical example.
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